They have similar meanings but serve a different purpose in a sentence. "Either" is an adjective or an adverb - it describes the condition of a noun or a verb or phrase in the sentence - usually describing one thing as being exclusive of another. "Whether" is more like a conjunction that joins two parts of a sentence together, both with a subject and a verb. Other common conjunctions are "but," "or," and "and." "Whether" is different because it does not always go between the two parts of the sentence it is joining, but can go at the beginning of the sentence. "Either the dog or the cat made a mess in the kitchen." is a fine sentence. "Whether the dog or the cat made a mess in the kitchen." is not a complete sentence. It is a fragment that needs another piece to make sense. It would set up the listener to expect some more information. "Whether the dog or the cat made the mess in the kitchen, I still have to clean it." "You decide whether you want to become a doctor or a lawyer" is an informal, conversational usage of "whether." It is really the joining of the two sentences "You decide that you want to become a doctor" and "You decide that you want to become a lawyer." The "whether" sets it up as a clear choice between the two. "You can decide to become either a doctor or a lawyer." would serve the same purpose. "There are knives on either side of the plate." does mean that both sides have knives. "There will be knives on either the left side or the right side of the plate." means that only one side will have knives. If "either" is used without "or" separating two alternatives, it usually means "both."

To me, either/or is usually used for objects/nouns and comparisons in direct statements while whether this/that is used for decision making and questions. For example: Either this coat or that one will work. One can either go to school or get a job. Whether you want to come over or stay home, the choice is yours. You decide whether you want to become a doctor or a lawyer.

Thank you! Your last comment tells me that there is a slightly difference between them, if I understood you right. Anyway, I've got to study more so that it gets clearer to me. Regards!

Also, in your statement about the restaurant either doesn't really mean "both" in the way that you are thinking. If you said "In a fancy restaurant both sides of your place setting..." you would imply that each side has the same utensils and the reader would visualize both sides at the same time. When you say "In a fancy restaurant either side of your place setting there are different knives and forks" you are painting a visual picture which tells the reader that each side is different and that he/she should look at one side then the other.